Introduction to the theory of topological insulators

One of the "hot topics" in condensed-matter theory in the last
decade has been the discovery and investigation of so-called
"topological insulators" (TIs).  Heuristically speaking, these
are materials whose ground-state electron wavefunctions wrap
around the Brillouin zone with a "twist" that cannot be removed
without closing the band gap.  In this talk, I will first give
an introduction to Berry phases and curvatures, mathematical
objects that are essential for describing the topological states
in question.  I will then give a brief survey of the different
types of topological states that are currently of such interest,
including quantum anomalous Hall (QAH) and quantum spin Hall (QSH)
systems in 2D, and strong and weak TIs in 3D.  Finally, I will
briefly review some of our own recent work in which we propose the
deposition of atoms with strong spin-orbit interactions onto the
surfaces of magnetic insulators as a promising route toward the
synthesis of QAH systems.  These hold promise to reproduce the
physics of the famous quantum Hall effect, but without the usual
constraints of low temperature and applied magnetic field.